Assume that \(X\cong PSU_{n}(q^{1/2})\) , with \(n \geqslant 3\) . Then \(X_{x}\)  lies in a large maximal
geometric subgroup \(Y\)  of \(X\)  by Theorem REF , and hence \(Y\)  is one of
the groups listed in [1]}, Proposition 4.17. Also \(\left(\Phi _{ef}^{\ast }(p),\left|X_{x}\right|\right) >1\) , and hence \(\left( \Phi _{ef}^{\ast }(p),\left|Y\right|\right) >1\) , where \(e=n\) 
or \(n-1\)  according as \(n\)  is odd or even respectively.
Assume that \(n\)  is odd. Filtering the list of subgroups in [1]},
Proposition 4.17, with respect to the property of \(\left( \Phi _{nf}^{\ast }(p),\left|Y\right|\right) >1\) , we see that the unique admissible
case is when \(Y\)  is a \(\mathcal {C}_{3}\) -subgroup of \(X\)  of type \(GU_{n/3}(3^{3})\) . Then \(Y\cong Z_{7}.PSU_{n/3}(3^{3}).Z_{(n/3,7)}.Z_{3}\)  by [3]}, Proposition 4.3.6(II). It
follows from Lemma REF (1) that \(\left|X\right|\leqslant \left|\mathrm {Out(}X\mathrm {)}\right|^{2}\cdot \left|Y\right|\cdot \left|Y\right|_{p^{\prime }}^{2}\)  which yields
inequality (6) in the proof of Proposition 4.3 of [4]} for \(t=q=3\) . Thus \((n/t,t)=(1,3)\) , and hence \(k^{2}=288\) , which is a
contradiction.
Assume that \(n\)  is even. Recall that \(n\geqslant 4\)  by Remark REF . The
constraint \(\left( \Phi _{(n-1)f}^{\ast }(p),\left|Y\right|\right)>1\)  reduces the list of the subgroups in [1]}, Proposition 4.17 to the
case \(Y\in \mathcal {C}_{1}(X)\) . Then \(Y\)  is the stabilizer in \(X\)  of a non-isotropic point of \(PG_{n-1}(q)\)  by [3]}, Propositions 4.1.4.(II) and 4.1.18.(II). Moreover, \(X_{x}=Y\)  by Lemma REF (1) and by [3]}, Proposition 4.1.4.(I). So, \(k^{2}=q^{(n-1)/2}\frac{q^{n/2}-1}{(q^{1/2}+1)}\) , which is exactly (REF ) in Lemma REF , with \(s=q^{1/2}\) . Hence, the same argument used in Lemma REF  rules out this case.
